Null geodesic congruences, gravitational lensing and CMB intensity profile

Abstract

It is known that the Cosmic Microwave Background (CMB) temperature fluctuations are modified by gravitational lensing since the angular positions of photons are altered by the metric perturbations. We reconsider this effect in the context of geodesic deviation on a perturbed Friedmann-Robertson-Walker (FRW) spacetime with scalar and tensor modes. We first give an alternative derivation of the Sachs-Wolfe effect by using the solutions of the null geodesic equation. Then, we determine the two-dimensional induced metric on the transverse cross section of the null geodesic congruence corresponding to a beam of photons after decoupling. This metric, whose variation along the congruence can be decomposed as expansion and shear, is shown to produce a nontrivial intensity profile which can be characterized by three variables analogous to the Stokes parameters of the polarization tensor.